Finite Group Databasebeta

← #28334875

Subgroups — #28334875

Order 13696.

234 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293161C211yesmaximal, characteristic
2C2×C4⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
3#20292359C212yesmaximal, complemented
4#20292356C212yesmaximal, complemented
5C4×M16×C107C211yesmaximal, characteristic, complemented
6C22×C107⋊C8—21no—
7#16812103—21no—
8#16812101C411yescharacteristic
9#16812097C411yescharacteristic
10#16812092—21no—
11#16812088C411yescharacteristic
12#16812083C411yescharacteristic
13C4×Dic214C2211yescharacteristic
14C4×D428C2211yescharacteristic
15C4×C22⋊D107—21no—
16#16812066—21no—
17C4⋊(C107⋊C8)—21no—
18C4⋊(C107⋊C8)C2212yes—
19C2×M16×C107—21no—
20C2×C42×C107C2211yescharacteristic
21#97×C107—21no—
22C4×C8×C107C2212yes—
23C2×Dic214C2×C411yescharacteristic
24C2×D428C2×C411yescharacteristic
25C2×C22⋊D107—21no—
26C2×C4×D107—21no—
27C2×C107⋊C8—22no—
28C2×C107⋊C8—42no—
29C4⋊Dic107C2×C411yescharacteristic
30C4⋊Dic107C2×C411yescharacteristic
31Dic107⋊C4—21no—
32Dic107⋊C4—21no—
33Dic107⋊C4—21no—
34Dic107⋊C4—21no—
35C22⋊Dic107—21no—
36(C2×D107)⋊C4—21no—
37(C2×D107)⋊C4—21no—
38(C2×D107)⋊C4—21no—
39(C2×D107)⋊C4—21no—
40C4×Dic107—21no—
41C22×C4×C107C2×C411yescharacteristic
42C22×C4×C107D411yescharacteristic
43C22×C4×C107D411yescharacteristic
44M16×C107—24no—
45C2×C8×C107—22no—
46C42×C107C2×C411yescharacteristic
47C42×C107D411yescharacteristic
48C42×C107D411yescharacteristic
49C42×C107C2311yescharacteristic
50C23×C107C22⋊C411yescharacteristic
51C22×D107—21no—
52C2×C4×C107—22no—
53C2×C4×C107—22no—
54C2×C4×C107—21no—
55C2×C4×C107—21no—
56C2×C4×C107C22⋊C411yescharacteristic
57C2×C4×C107C22⋊C411yescharacteristic
58C2×C4×C107C22⋊C411yescharacteristic
59C2×C4×C107C22×C411yescharacteristic
60C2×C4×C107C2×D411yescharacteristic
61C2×C4×C107C2×D411yescharacteristic
62C22⋊D107—42no—
63C2×Dic107—21no—
64C2×Dic107—21no—
65C2×Dic107—21no—
66D428—22no—
67C4×D107—41no—
68C4×D107—41no—
69Dic214—22no—
70C8×C107—24no—
71C107⋊C8—42no—
72C22×C107—21no—
73C22×C107—21no—
74C22×C107#10412yes—
75C22×C107C2×C22⋊C411yescharacteristic
76C2×D107—41no—
77C2×D107—41no—
78C2×D107—41no—
79C4×C107—21no—
80C4×C107—21no—
81C4×C107—21no—
82C4×C107—21no—
83C4×C107#13112yes—
84C4×C107AGL(1,ℤ/8)11yescharacteristic
85C4×C107#13711yescharacteristic
86Dic107—41no—
87Dic107—41no—
88Dic107—41no—
89C2×C107—22no—
90C2×C107#40811yescharacteristic
91C2×C107#41711yescharacteristic
92C2×C107C2×#10411yescharacteristic
93D107—81no—
94#1483—1071nomaximal, Sylow
95C107#148311yescharacteristic, complemented, Sylow
96#529—1071no—
97C2×C4⋊C8—1071no—
98C4×M16D10711yescharacteristic, complemented
99Q8⋊C8—1072no—
100D4⋊C8—1072no—
101C2×M16—21no—
102C22×C8—2141no—
103#126—2141no—
104#125—1071no—
105#124—1071no—
106#123—2141no—
107C22⋊Q8—1071no—
108#121—1071no—
109C4×Q8—1071no—
110C4×D4—1071no—
111C4×D4—2141no—
112#117—2141no—
113C2×C42C2×D10711yescharacteristic
114C4⋊C8—1072no—
115C4⋊C8—2141no—
116#97—21no—
117C4×C8C2×D10712yes—
118C2×Q8—1071no—
119C2×D4—1071no—
120C2×D4—2141no—
121C22×C4—2141no—
122C22×C4C4×D10711yescharacteristic
123C22×C4D42811yescharacteristic
124C22×C4C22⋊D10711yescharacteristic
125M16—24no—
126C2×C8—22no—
127C2×C8—4282no—
128C2×C8—2142no—
129C4⋊C4—1071no—
130C4⋊C4—1071no—
131C4⋊C4—2141no—
132C4⋊C4—2141no—
133C4⋊C4—2141no—
134C4⋊C4—2141no—
135C22⋊C4—2141no—
136C22⋊C4—2141no—
137C22⋊C4—2141no—
138C22⋊C4—2141no—
139C22⋊C4—2141no—
140C42—2141no—
141C42C4×D10711yescharacteristic
142C42D42811yescharacteristic
143C42C22⋊D10711yescharacteristic
144C42C22×D10711yescharacteristic
145C23—2141no—
146C23(C2×D107)⋊C411yescharacteristic
147Q8—2142no—
148D4—4282no—
149D4—2142no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—4281no—
155C2×C4—4281no—
156C2×C4—2141no—
157C2×C4—2141no—
158C2×C4—2141no—
159C2×C4(C2×D107)⋊C411yescharacteristic
160C2×C4(C2×D107)⋊C411yescharacteristic
161C2×C4(C2×D107)⋊C411yescharacteristic
162C2×C4C2×C4×D10711yescharacteristic
163C2×C4C2×C22⋊D10711yescharacteristic
164C2×C4C2×D42811yescharacteristic
165C8—24no—
166C8—4282no—
167C22—21no—
168C22—21no—
169C22—4281no—
170C22—4281no—
171C22—4281no—
172C22#1681203712yes—
173C22C2×(C2×D107)⋊C411yescharacteristic
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—4281no—
179C4—4281no—
180C4—4281no—
181C4#1681211712yes—
182C4#1681213711yescharacteristic
183C4#1681214211yescharacteristic
184C2—22no—
185C2—8561no—
186C2#2029259511yescharacteristic
187C2#2029262911yescharacteristic
188C2C2×#1681203711yescharacteristic
subgroupG/H
centerC2×C4C2×D428
commutatorC2×C4×C107C22×C4
FrattiniC42C22×D107
FittingC4×M16×C107C2
socleC22×C107C2×C22⋊C4
prime radical#16812083C4
cosocleC42×C107C23
hypercenterC4×M16D107
hypocenterC107#1483
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4⋊(C107⋊C8)C2semidirect
3. #20292359C2semidirect
4. #20292356C2semidirect
5. C4×M16×C107C2semidirect
95. C107#1483semidirect
98. C4×M16D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202931612
C2×C4⋊(C107⋊C8)2
#202923592
#202923562
C4×M16×C1072
#1483107